Triangles win?

0:00Points in the plane

Scatter points across the plane, one per unit of area on average, and give every pair an energy that depends only on the distance between them. Which arrangement has the least energy per point? The natural candidate is the triangular lattice, the pattern of equilateral triangles. The papers here allow any potential that is completely monotone in the squared distance: never negative, never increasing, convex, with derivatives alternating in sign forever. Every Gaussian and every inverse power qualifies. In our computation with the Gaussian e^(−π × squared distance), the triangular lattice has energy 0.160 per point, the square lattice 0.180, and the honeycomb, with two points in each cell, 0.272. Among lattices alone, the answer was classical: Rankin, Cassels, Ennola and Diananda settled the inverse powers whose sums converge, and Montgomery, in 1988, every Gaussian, so among lattice shapes the triangle sits at the bottom. But a competitor need not be a lattice. Cohn and Kumar conjectured in 2007 that sharp Fourier bounds would prove the triangular lattice universally optimal, along with the E8 and Leech lattices, and Cohn, Kumar, Miller, Radchenko and Viazovska later proved those two higher-dimensional cases. In the plane, only restricted competitors were handled: some periodic families, and small displacements of the lattice. In our search, 16 random points on a torus that fits the triangular pattern always slid down to its energy, while on a square torus the best we found was 0.165. Two neighboring problems point the same way. Equal disks packed in the triangular pattern cover π/(2√3), about 90.7 percent of the plane, and nothing does better. Cohn and Elkies gave a Fourier bound for packings and conjectured that, in dimensions 2, 8 and 24, one auxiliary function makes it exact; Viazovska found one in dimension 8, and with collaborators in 24. Our own linear program with 20 Laguerre–Gaussian functions comes within seven millionths of the planar density, but a numerical search only approaches the bound. Then there are charges. Sandier and Serfaty, studying vortices in superconductors, conjectured that the triangular lattice minimizes a renormalized energy for logarithmic repulsion against a uniform background, and Brauchart, Hardin and Saff conjectured the constant in the linear term of the least logarithmic energy of points on a sphere. Our best local minima give −0.048 at 50 points and −0.0525 at 800, drifting toward their value, −0.0556.

2:58The claims

Result 090 is four OpenAI preprints. The first, 70 pages, dated September 23, proves universal optimality in the plane: for every completely monotone potential of squared distance, and every configuration whose density in growing centered disks tends to one, the energy per point, in the lower limit, is at least the triangular lattice’s, even when the energies are infinite. The second, 31 pages, dated September 26, proves the same theorem with a smaller, independent construction. The third, 49 pages, proves the Sandier–Serfaty conjecture and, with a formula of Bétermin and Sandier, the Brauchart–Hardin–Saff constant. The fourth, also 49 pages, builds an exact Cohn–Elkies function in the plane. The first paper also proves triangular minimality for renormalized Riesz energies with exponent between zero and two, where the raw sums diverge even for the lattice. The claims identify minimum values, not every minimizer, and Cohn and Kumar’s stronger request, a sharp Fourier function for each potential, is not claimed. OpenAI published Lean formalizations of the energy theorem, the atomic paper’s Gaussian functions and the packing certificate: about 69,000 lines for energy, and over 900,000 for packing, mostly interval data checked by Lean’s kernel. The Coulomb paper and the Riesz results are not formalized. We have not run the checks.

4:31How the proofs go

Both energy papers reduce everything to Gaussians. For each Gaussian they build a radial function f that stays below it, whose Fourier transform is never negative, that touches the Gaussian at every nonzero lattice point, and whose transform vanishes at every nonzero point of the dual lattice, which for the triangular lattice is the same lattice turned a quarter turn. For any configuration of density one, the two sign conditions bound the energy below by f̂(0) − f(0), and Poisson summation over the lattice shows this number is exactly the lattice energy. Finally, every completely monotone potential, shifted slightly, is a positive mixture of Gaussians and a constant, by a classical theorem of Bernstein and Widder, so the Gaussian inequalities add up to the full theorem. The hard part is building f for every Gaussian at once. Rescaled squared lengths of triangular lattice vectors are the numbers j² + jk + k², which leave only certain remainders on division by 12: 0, 1, 3, 4, 7 or 9. The 70-page paper puts a node at every such positive integer, uses a product of squared sines with a double zero at each node, and builds columns that prescribe a value and a slope at every node. Finite blocks of 168 coordinates are certified with interval arithmetic, the infinite tail is solved because its couplings are small, and the signs between nodes are certified with Bernstein polynomial bounds over whole ranges of the Gaussian parameter. The atomic paper uses nodes modulo 36 and blocks of 20. The packing paper built this machinery first, for one function. The Coulomb paper takes a direct route, with no universal-optimality input. On a square torus, minimizing points stay a definite distance apart. Their Voronoi cells, the regions nearest each point, split into sectors, one per edge, and on average a point has at most 6. With one trial potential, glued continuously across all sectors, each sector’s contribution is at least a negative constant plus fixed multiples of its area and angle. Areas and angles add up exactly, so at most 6 sectors per point gives a bound calibrated to equal the triangular value. Two scalar inequalities behind this are verified with interval arithmetic. Sandier and Serfaty’s periodic approximation carries the bound to the whole plane, and Bétermin and Sandier had shown that the sphere constant takes the conjectured value exactly when the triangular lattice wins.

7:11Where it stands

None of the four preprints is peer reviewed, and none has been revised. OpenAI’s Lean formalizations cover the energy theorem, the atomic paper’s Gaussian functions and the packing certificate; we have not run them. The Coulomb paper and the sphere constant rest on that paper and its interval arithmetic, and the Riesz step is unformalized analysis. The packing paper mentions a 2026 thesis abstract announcing a related function built from modular forms. None of the papers is among those OSVerify reviewed, and we have seen no expert review. If they hold, the plane largely joins dimensions 8 and 24: no configuration of density one beats the triangular lattice, for any completely monotone potential; it minimizes the Coulomb and long-range Riesz renormalized energies, fixes the Brauchart–Hardin–Saff constant, and has an exact Cohn–Elkies function. Whether the minimizers are unique, and whether every single potential has its own sharp Fourier function, are not settled here.